Description: From canth27030, we know that , but we
cannot prove that (this is the
Continuum
Hypothesis), nor can we prove that it is less than any bound whatsoever
(i.e. the statement is consistent for
any ordinal ).
However, we can prove that is not
equal to , nor , on
cofinality grounds, because by Konig's Theorem konigth8207 (in the form of
cfpwsdom8222), has uncountable cofinality, which
eliminates limit alephs like . (The first limit
aleph that is not eliminated is ,
which has cofinality .) (Contributed by Mario
Carneiro, 21-Mar-2013.)